1. Internet Engineering Czesław Smutnicki Discrete Mathematics – Discrete Optimization

2. CONTENT S Numerical troubles Packages Tools Useful methods

3. FIND EXTREMES OF THE FUNCTION 2D 1D DE JONG TEST FUNCTION OPTIMIZATION TROUBLES. NICE BEGINNINGS OF BAD NEWS

4. GRIEWANGK TEST FUNCTION FIND EXTREMES OF THE FUNCTION 2D OPTIMIZATION TROUBLES. MULTIPLE EXTREMES

5. LANGERMANN TEST FUNCTION FIND EXTREMES OF THE FUNCTION 2D OPTIMIZATION TROUBLES. EXPONENTAL NUMBER OF EXTREMES

6. FOX HOLES TEST FUNCTION FIND EXTREMES OF THE FUNCTION 2D OPTIMIZATION TROUBLES. DECEPTION POINTS

7.  Please wait. Calculations will last 3 289 years INSTANCE FROM PRACTICE ! ! ? NONLINEAR FUNCTION OF 1980 VARIABLES !!! CURSE OF DIMENSIONALITY OPTIMIZATION TROUBLES. TIME OF CALCULATIONS/COST OF CALCULATION S LAB INSTANCE 5..20 VARIABLES NP- HARDNESS

8. SOLUTION SPACE The smallest practical instance FT10 of the job-shop scheduling problem (waited 25 years for the solving), consists of 10 jobs, 10 machines, 100 operations; solution space contains 10 48 discrete feasible solutions; each solution has dimension 90; the greatest currently used benchmarks have dimension 1980 dimension and size FT 10 corresponds to printed area of 10 32 km 2 (Jupiter has 10 10 km 2 ) if single solution is a dot 0.01 x 0.01 mm OPTIMIZATION TROUBLES. SIZE OF THE SOLUTION SPACE

9. Example: job-shop scheduling problem; relative Hamming distances DIST between a feasible solution and the „best” solution are distributed normally in the solution space Goal function values are distributed normally in the solution space; OPTIMIZATION TROUBLES. DISTRIBUTION OF THE GOAL FUNCTION VALUES

10. Example: job-shop scheduling problem SIMULATION OF GOAL FUNCTION VALUES TOWARDS CENTER OF THE SPACE OPTIMIZATION TROUBLES. FUR

11. OPTIMIZATION TROUBLES. ZOOM IN ON THE FUR Example: job-shop scheduling problem SIMULATION OF GOAL FUNCTION VALUES TOWARDS CENTER (ZOOM)

12. Transformation of a sample of random solutions from the 90D space into 2D space. OPTIMIZATION TROUBLES. STONE FOREST

13. PROPERTIES OF SOLUTION SPACE LANDSCAPE BIG VALLEY – positive correlation between goal function value and the distance to optimal solution (the best found solution); in the big valley the concentration of local extremes is high. The size of the valley is usually relatively small in relation to the size of the whole solution space. RUGGEDNESS – measure of diversity of goal function values of related (neighboring) solutions; rruggedness is greater if diversity of the goal function value in the neighborhood of this point is greater; less differentiation of the goal function value means the flat landscape. THE NUMBER OF LOCAL EXTREMES (peaks) in relation to to the size of the solution space DISTRIBUTION OF LOCAL EXTREMES experimental OTHER MEASURES autocorrelation function, correlation function between random trajectories, landscape statistically isotropic, fractal landscape, correlation between genes (epitasis), correlation of the distance of fitness

14. CURRENT STATE IN DISCRETE OPTIMIZATION Packages and solvers (LINDO, CPLEX, ILOG, …) Exact methods (B&B, DP, ILP, BLP, MILP, SUB,…) Approximate methods (…): heuristics, metaheuristics, meta 2 heuristics Quality measures of approximation (absolute, relative, …) Analysis of quality measure (worst-case, probabilistic, experimental) Calculation cost (pessimistic, average, experimentally tested) Approximation schemes (AS, polynomial-time PTAS, fully polynomial-time FPTAS) Inapproximality Useful experimental methods (…) „No free lunch” theorem Public benchmarks Parallel and distributed methods: new class of algorithms

15. OPTIMIZATION HISTORY/TRENDS Priority rules Theory of NP-completeness Plynomial-time algorithms Exact methods (B&B, DP, ILP, BLP,…) Approximation methods: quality analysis Approximation schemes (AS, PTAS, FPTAS, …) Inapproximality theory Competitive analysis (on-line algorithms) Metaheuristics Theoretical f o undations of metaheuristics Parallel metahuristics Theoretical f o undations of parallel metaheuristics

16. constructive/improvement priority rules random search greedy randomized adaptive simulated annealing simulated jumping estimation of distribution tabu search adaptive memory search variable neighborhood search evolutionary, genetic search differential evolution biochemistry methods immunological methods ant colony optimization particle swarm optimization neural networks threshold accepting path search beam search scatter search harmony search path relinging adaptive search constraint satisfaction descending, hill climbing multi-agent memetic search bee search intelligent water drops * * * * * METHODS RESISTANT TO LOCAL EXTREMES APPROXIMATE METHODS

17. EVOLUTION : DARWIN ’S VIEW. GENETIC ALGORITHMS individual, gen e, chromosome, trait population (structure, size, composition) crossing-over (what is the key of progress?) mutation (insurance?) sex ? democracy/elitarism theoretical properties individual=solution=genotype≠fenotype GOAL OF THE NATURE? optimization, fitness, continuity preservation, follow up changes SUCCESION: genetic material carrie s data for body construction EVOLUTION: crossing over, mutation SELECTION: soft/hard

18. EVOLUTION : DARWIN ’S VIEW. COMPONENTS SOLUTION FEASIBILITY SELECTION SCHEME CROSSING OVER MATTING POOL CHROMOSOM MUTATION OPERATOR MSXF BIG VALLEY PHENOMENON GENOTYPE FENOTYPEREPAIRING CONTROL OF POPULATION DYNAMICS INTENSIFICATION MORE … GENE EXPRESSION CODING LETHALITY

19. EVOLUTION : DARWIN ’S VIEW. COPYING FROM THE NATURE control of population dynamics/preserving diversi t y parents matching strategies: (sharing function to prevent to o close relative parents; incest preventing by using Hamming distance to evaluate genotype similarity) structures of the population (migration, diffusion models) social behavior patterns (satisfied, glad, disappointed -> clonning, crossing- over, mutation) adaptive mutation gene expression distributed populations …

20. EVOLUTION : DARWIN ’S VIEW. MULTISTEP FUSION MSXF SOURCE SOLUTION (PARENT) NEIG H BORHOOD OF THE SOURCE TARGET SOLUTION (PARENT) TA R GET NEIGHBORHOOD TRAJECTORY = GOAL ORI E NTED PATH SUCCESSIVE NEIGHBOURHOODS SEARCHED IN THE STOCHASTIC WAY DEPENDING THE DISCTANCE TO TARGET DISTANCE TO TARGET

21. EVOLUTION : LAMARCK/BALDWIN ’S VIEW. MEMETIC ALGORITHMS GOAL OF THE NATURE? optimization, fitness, continuity preservation, follow up changes, transfer knowledge to successors SUCCESION: genetic material carries data for body building plus acquired knowledge EVOLUTION: crossing over, mutation, learning SELECTION: soft/hard individual, mem e, chromosome, trait population (structure, size, composition, learning) crossing-over, mutation, learning theoretical properties ? individual=solution=memotype≠fenotype

22. DIFFERENTIAL EVOLUTION Differential evolution is a subclass of genetic search methods. Democracy in creating successors with using crossover and mutation in GS has been replaced in DE by directed changes to fathom solution space. DE starts from the random population of individuals (solutions). In each iteration something similar to mutation and crossover is performed, however in completely different way than in GS. For each solution x from the space, an offspring y is generated as the trial solution being the extension of a selected random solution a and two directional solutions b and c (analogy to parents) selected at random. Generation is based on linear combination with some random parameters. Separate mechanism prevents generating an offspring by simple copying of the parent. Significant role plays the mutation, which due to specific strategy, is self-adaptive and goal-oriented with respect to the direction, scale and range. If the trial solution is better, it is accepted; otherwise it is released. Iterations are repeated until the fixed a priori number of iterations has been reached, or stagnation has been detected. The method owns some specific tuned parameters: differential weight, crossover probability, … selected experimentally.

23. ARTIFICIAL IMMUNE SYSTEM Antigen (invasive protein) represents new problem to solve or new (or temporary) constraints set for the solution of already solved problem. Variety of possible antigens is huge, frequently infinite. Moreover, sequence of presented antigens is not known a priori. Antibody (protein blocking antigent, directed against intruder) corresponds to an algorithm which produces a solution to the problem. Variety of antibodies is usually small, however mechamisms exist of their aggregation and recombination in order to produce new antibodies with various properties. Patterns of antibodies are collected in the library, which constitutes memory of the system. Matching (fitness) is the selection of antibody for the an t igen. Matching is ideal, if the antibody allow us to generate solution of the problem which is globally optimal under given constraints. Otherwise, certain defined measure is used to evaluate quality of the maching. Bad maching forces the system to seek for new types of antibodies, usually by using evolution. antigen = problem or instance antibody = solution LIBRARY OF ANTIBODIES fitness recombination

24. ANT SEARCH. COOPERATIVE SWARMS ANT seeks for food leaves pheromone on the trail moves at random, but prefers pheromone trails pheromone density decreases in time control system pheromone generator moving drive Pheromone detectors

25. ANT SEARCH. SEEKING FOODS. DISCOVERING THE PATH A E B C H E A DD B C H E A D B C H E A

26. ANT SEARCH. PHEROMONE DISTRIBUTION

27. PARTICLE SWARM OPTIMIZATION swarm is a large set of individuals (particles) moving together each individual performs the search trajectory in the solution space trajectories are distributed, correlated and take into account experiences of individuals location of the individual (solution) is described by the location vector x, changes of location is described by velocity vector v velocity equation contain t s an inerti A term and two directional terms weighted by using some random parameters location of the individual depends on : recent (previous) position, experience (best location up to now), location of the leader of the swarm, the best up to now solution form the most promising direction of the search

28. BEE SEARCH Neighborhood search combined with random search and supported by cooperation (learning). bee swarm collects honey in hive each bee performs the random path (solution) to the search region of nectar selected elite bees in hive perform „waggle dance” in order to inform other bees about promising search regions (direction, distance, quality) flowers & nectar hive bee waggle dance = distribution of knowledge bee trajectory = solution visited site = neighborhood nectar amount = goal function

29. TABU SEARCH human thinking in the process of seeking a solution the method „best in local neighborhood” repeated from the best recently found forbidding the return to solutions already visited to prevent cyclic (wandering around); short term memory NEIGHBOURHOOD SUCCESSIVE NEIGHBOURHOODS EXPLORED EXHAUSTIVELY STARTING SOLUTION

30. ADAPTIVE MEMORY SEARCH gathering data in human brain during the process of seeking a solution the method „best” in the current heighbourhood (a few solution relatively close to the current) repetition from the best recently found; intensification of the search operational (short term) memory: prohibition of coming back to solutions already visited to prevent wandering tactic memory: set direction of the search strategic memory: selection of search regions (basins of attraction); diversification recency based, frequency based memory

31. INTELLIGENT WATER DROPS Based on the dynamic of the river systems, action and reaction, that happen among water drops in rivers: a drop has some (static ) parameters, namely velocity, soil; these parameters may change during the lifetime (e.g. iterative cost) drops flow from a source to destination a drop starts with some initial velocity and zero soil during the flow, drop removes some soil from the environment speed of the drop incereases non-linearly inverse ly to the amount of soil; path with less soil is faster than path with more soil soil is gathered in the drop and removed from the environment drop statistically prefers path with lower soil

32. annealing = slow cooling of ferromagnetic or antyferromagnetic solid in order to eliminate internal stretches Boltzman (harmonic) Logar i t h mic (Hajek lemay) Geometric SIMULATED ANNEALING. COOLING SCHEMES

33. Random starting solution Sequence of k trial moves in the space K steps in each fixed temperature Starting temperature adjusted automatically Adaptive speed of cooling SIMULATED ANNEALING. AUTOTUNING p  0.9

34. SIMULATED JUMPING annealing by successive heating and cooling, in order to eliminate internal stretches of the spin-glass solid (mixed ferromagnetic and antyferromagnetic material); the aim is to penetrate high barriers that exist between domains

35. DISTANCE MEASURES IN THE SOLUTION SPACE complexity variance mean receipt number of inversion in  -1 o  n minus the number of cycles in  -1 o  n minus the lenght of the maximal increasing subsequence in  -1 o  measure D A ( ,  ) D S ( ,  ) D I ( ,  ) Move type AS I DISCRETE OPTIMIZATION. SOLUTION SPACE PROPERTIES

36. There exists strong correlation between quality of the function value (RE) and distance to the best solution (DIST); this correlation is preserved after transformation of the solution to x/y coordinates BIG VALLEY startbest SELECTED INSTANCES. BIG VALLEY

37. startbest SELECTED METHODS. RANDOM SEARCH RANDOM SEARCH TRAJECTORY Random search offers slow convergence to the good solution because it doesn’t use any information about structure of the solution space

38. startbest Simulated annealing offers moderate speed of convergence to the good solution; it is much more similar to the random search than to goal-oriented search SELECTED METHODS. SIMULATED ANNEALING SIMULATED ANNEALING TRAJECTORY

39. startbest TABU SEARCH TRAJECTORY Tabu search offers quick convergence to the good solution; this is the fast descent method supported by adaptive memory SELECTED METHODS. TABU SEARCH

40. PARALLEL OPTIMIZATION: NEW CLASS OF ALGORITHMS Theoretical models of parallel calculation: SISD, SIMD, MISD, MIMD Theoretical models of memory access: EREW, CREW, CRCW Parallel calculation environments: hardware, software, GPGPU Shared memory programming: Pthreads (C), Java threads, Open MP (FORTRAN, C, C++) Distributed memory programing, message-passing, object-based, Internet computing: PVM, MPI, Sockets, Java RMI, CORBA, Globus, Condor Measures of quality of parallel algorithms: runtime, speedup, effciency, cost Single/multiple searching threads; granularity Independent/cooperative search threads Distributed (reliable) calculations in the net

41. PARALLEL OPTIMIZATION: FESTIVAL OF APPROACHES SIMULATED ANNEALING: – Single thread, conventional SA, parallel calculation of the goal function value; fine grain; theory of convergence – Single thread, pSA, parallel moves, subset of random trial solutions selected in the neighborhood, parallel evaluation of trial solutions; theory of convergence – Exploration of equilibrium state at fixed temperature in parallel – Multiple independent threads; coarse grain – Multiple cooperative threads; coarse grain GENETIC SEARCH: – Single thread, conventional GA, parallel calculation of the goal function value; small grain; theory of convergence – Single thread, parallel evaluation of population; – Multiple independent threads; coarse grain – Multiple cooperative threads, distributed subpopulations: migration, diffusion, island models – …

42. Thank you for your attention DISCRETE MATHEMATICS Czesław Smutnicki